Periodic functions

Where \(k\in\mathbb{z}\) and \(T\in\mathbb{R}\)

Sum of Two Periodic Signals

If \(\( x_1(t) \)\) and \(\( x_2(t) \)\) are periodic with periods \(\( T_1 \)\) and \(\( T_2 \)\):

\[ x_1(t + T_1) = x_1(t), \quad x_2(t + T_2) = x_2(t) \]

The sum of both signals is:

\[ x(t) = x_1(t) + x_2(t) \]

Condition for the Periodicity of the Sum

For \(\( x(t) \)\) to be periodic, there must exist a common period \(\( T \)\) such that:

\[ T = k_1 T_1 = k_2 T_2 \]

where ( k_1, k_2 ) are positive integers.

Common Period and Least Common Multiple

The smallest common period is the least common multiple (lcm) of \(\( T_1 \)\) and \(\( T_2 \)\):

\[ T = \operatorname{lcm}(T_1, T_2) \]

If the ratio of the periods is a rational number:

\[ \frac{T_1}{T_2} \in \mathbb{Q} \]

Then, the sum \(\( x_1(t) + x_2(t) \)\) will be periodic.

If the ratio is irrational, the resulting signal will not be periodic.

Example